3. Observations and Problems¶
3.1 Speed and Error of Monte Carlo Simulations¶
The Monte Carlo technique is a very powerful tool for pricing options, especially in higher dimensions, however it is computationally expensive and due to reliance on generating a finite number of realisations using random numbers, the option price is subject to error and will vary with each run of the simulation.
As discussed in $\text{Section 1.4}$, the error in the option price is governed by the greater of the error due to the discreteness of the time step or the error due to the finite number of paths. This is given by $\text{Equation (24)}$. The error in the option price is therefore proportional to the square root of the number of paths, $N$, and inversely proportional to the time step, $\delta t$. This means that the error in the option price can be reduced by increasing the number of paths, $N$, and decreasing the time step, $\delta t$. However, this comes at the cost of increased computational time. In other words to improve the accuracy of the option price by a factor of 10 we must perform 100 times as many simulations.
In the follow section we will consider how variance reduction techniques improves the speed and standard error of the Monte Carlo simulations.
3.2 Variance Reduction with Antithetic Variates¶
We are able to speed up the convergence of the Monte Carlo simulations by using variance reduction techniques as oppose to increasing the number of simulations. The two most common variance reduction techniques are antithetic variates and control variates techniques. We will focus on the antithetic variates technique in this report.
In the antithetic variates technique we calculate two estimates for the option value, one using the original set of Normal random numbers $\phi_{i}$ and one using the negative of the original set of Normal random numbers $-\phi_{i}$. We then take the average of the two estimates to give the final estimate of the option value. This works due to the symmetry of the Normal distribution and works on the same principle as a hedged portfolio having a smaller variance than that of an unhedged portfolio.
Imagining a portfolio consisting of an option written on an asset $S_{1}$ and an option written on asset $S_{2}$, which is perfectly negatively correlated with $S_{1}$ and which has the same price as $S_{1}$. Then the risk-neutral random walks of the two assets are given by the SDEs:
$$
d S_{1,t}=r S_{1,t} d t+\sigma S_{1,t} d W_{t} \tag{26}
$$$$
d S_{2,t}=r S_{2,t} d t-\sigma S_{2,t} d W_{t} \tag{27}
$$
The value of the two option are identical as the price and volatility of the two assets are the same. However, the variance of the payoff of the portfolio containing both options is considerably smaller than the variance of the payoff of each individual option. This is because the two options are perfectly negatively correlated. When we calculate the average of the option payoffs, the fluctuations in the realisation of one path ($S_{1,t}$) are offset by the fluctuations in its antithetic variate ($S_{2,t}$). The result of this is a smaller spread of estimates around the expected value of the option price.
3.2.1 Antithetic Variates in the Euler-Maruyama Scheme¶
In the Euler-Maruyama scheme we can use antithetic variates to generate two paths for the underlying asset price, one using $\phi_{i}$ and one using $-\phi_{i}$. We can then use the average of the two paths payoffs to calculate the option price.
The risk-neutral asset price, $S_{1,t}$ and its antithetic variate $S_{2,t}$ using the Euler-Maruyama method over a time step $\delta t$ are given as:
$$
S_{1,t+\delta t}=S_{1,t} \exp \left(\left(r-\frac{1}{2} \sigma^{2}\right) \delta t+\sigma \sqrt{\delta} t \phi_{i}\right) \tag{28}
$$$$
S_{2,t+\delta t}=S_{2,t} \exp \left(\left(r-\frac{1}{2} \sigma^{2}\right) \delta t+\sigma \sqrt{\delta} t (-\phi_{i})\right) \tag{29}
$$
3.2.2 Antithetic Variates with Asian Options¶
Considering the pricing of fixed strike Asian options with time to expiry, $T - t$, we can write the expected present value of a fixed strike arithmetic Asian call, $C$ and its antithetic variate, $\bar{C}$, option using $\text{Equation (12)}$ and $\text{Equation (14)}$ as:
$$
C = e^{-r(T-t)} \mathbb{E}^{\mathbb{Q}}\left[\max \left( \frac{1}{N} \sum^{N}_{i=1} S_{1}(t_i) - E, 0 \right) \right] \tag{30}
$$$$
\bar{C} = e^{-r(T-t)} \mathbb{E}^{\mathbb{Q}}\left[\max \left( \frac{1}{N} \sum^{N}_{i=1} S_{2}(t_i) - E, 0 \right) \right] \tag{31}
$$
Similarly for the fixed strike arithmetic Asian put, $P$ and its antithetic variate, $\bar{P}$, option using $\text{Equation (12)}$ and $\text{Equation (15)}$ as:
$$
P = e^{-r(T-t)} \mathbb{E}^{\mathbb{Q}}\left[\max \left( E - \frac{1}{N} \sum^{N}_{i=1} S_{1}(t_i), 0 \right) \right] \tag{32}
$$$$
\bar{P} = e^{-r(T-t)} \mathbb{E}^{\mathbb{Q}}\left[\max \left( E - \frac{1}{N} \sum^{N}_{i=1} S_{2}(t_i), 0 \right) \right] \tag{33}
$$
where $S_{1}(t_i)$ and $S_{2}(t_i)$ are the asset prices at time $t_{i}$ for the original and antithetic variate paths respectively and $\mathbb{E}^{\mathbb{Q}}$ is the expectation under the risk-neutral measure.
3.2.3 Antithetic Variates with Lookback Options¶
Considering the pricing of fixed strike maximum and minimum lookback options with time to expiry, $T - t$, we can write the expected present value of a fixed strike maximum lookback call, $C$ and its antithetic variate, $\bar{C}$, option using $\text{Equation (20)}$ and $\text{Equation (22)}$ as:
$$
C = e^{-r(T-t)} \mathbb{E}^{\mathbb{Q}}\left[\max \left( \max \left( S_{1}(t_1), S_{1}(t_2), \dots, S_{1}(t_N) \right) - E, 0 \right) \right] \tag{34}
$$$$
\bar{C} = e^{-r(T-t)} \mathbb{E}^{\mathbb{Q}}\left[\max \left( \max \left( S_{2}(t_1), S_{2}(t_2), \dots, S_{2}(t_N) \right) - E, 0 \right) \right] \tag{35}
$$
Similarly for the fixed strike maximum lookback put, $P$ and its antithetic variate, $\bar{P}$, option using $\text{Equation (21)}$ and $\text{Equation (23)}$ as:
$$
P = e^{-r(T-t)} \mathbb{E}^{\mathbb{Q}}\left[\max \left( E - \max \left( S_{1}(t_1), S_{1}(t_2), \dots, S_{1}(t_N) \right), 0 \right) \right] \tag{36}
$$$$
\bar{P} = e^{-r(T-t)} \mathbb{E}^{\mathbb{Q}}\left[\max \left( E - \max \left( S_{2}(t_1), S_{2}(t_2), \dots, S_{2}(t_N) \right), 0 \right) \right] \tag{37}
$$
where $S_{1}(t_i)$ and $S_{2}(t_i)$ are the asset prices at time $t_{i}$ for the original and antithetic variate paths respectively and $\mathbb{E}^{\mathbb{Q}}$ is the expectation under the risk-neutral measure. The same equations can be used for the fixed strike minimum lookback options by replacing the maximum with a minimum.
3.2.4 Antithetic Variates Implementation¶
We will now implement the antithetic variates technique by creating a new AntitheticMonteCarloSimulation class. We will add the method run_Antithetic_GBM() which will behave similarly to run_GBM but instead simulate the underlying asset price paths using the Euler-Maruyama scheme with antithetic variates, it will be called using the same generate_Paths() method to maintain the same function signature. We will also create new classes for each of the options to inherit from this AntitheticMonteCarloSimulation class and implement their adjusted payoff calculation with the same function signature of call_Price and put_Price. This means we can create instances of the old and new classes and run their methods in the same way.
To test the implementation of the antithetic variates technique we will use the same sample data as before but run only half the number of total Monte Carlo simulations. We'll then compare the option prices and standard errors of the original and antithetic variate classes.
